#299781

order	512
nilpotency class	3
isoclinism	isoclinic class of #299765
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

797 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 797 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,13)(5,11)(6,10)(8,14)(9,16)(12,15)(17,18,23,28)(19,25,31,40)(20,32,36,47)(21,37,22,27)(24,38,41,30)(26,35,44,34)(29,45,33,39)(42,48,43,46);(1,3,6,8)(2,7,10,14)(4,9,15,11)(5,13,16,12)(17,19,22,38,23,31,21,30)(18,24,37,40,28,41,27,25)(20,29,35,48,36,33,34,46)(26,39,47,43,44,45,32,42);(1,4,5,14,6,15,16,7)(2,8,9,13,10,3,11,12)(17,19)(18,25)(20,33)(21,30)(22,38)(23,31)(24,37)(26,43)(27,41)(28,40)(29,36)(32,39)(34,48)(35,46)(42,44)(45,47);(17,20,22,35,23,36,21,34)(18,26,37,47,28,44,27,32)(19,29,38,48,31,33,30,46)(24,39,40,43,41,45,25,42)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=f,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,g^c=ghi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206977);